378 INTEGRAL CALCULUS

At the points of intersection the co-ordinates of the

curves are equal. Sincey=x^2 theny^2 =x^4. Hence

equating they^2 values at the points of intersection:

x^4 = 8 x

from which, x^4 − 8 x= 0

and x(x^3 −8)= 0

Hence, at the points of intersection,x=0 andx=2.

Whenx=0,y=0 and whenx=2,y=4. The

points of intersection of the curves y=x^2 and

y^2 = 8 xare therefore at (0,0) and (2,4). A sketch is

shown in Fig. 38.8. Ify^2 = 8 xtheny=

√

8 x.

Shaded area

=

∫ 2

0

(√

8 x−x^2

)

dx=

∫ 2

0

(√

8

)

x

1

(^2) −x^2
)
dx

⎡
⎣
(√
8
)x^32
3
2
−
x^3
3
⎤
⎦
2
0

{√
8
√
8
3
2
−
8
3
}
−{ 0 }

16
3
−
8
3

8
3
= 2
2
3
square units
Figure 38.8
The volume produced by revolving the shaded area
about thex-axis is given by:
{(volume produced by revolvingy^2 = 8 x)
−(volume produced by revolvingy=x^2 )}
i.e.volume=
∫ 2
0
π(8x)dx−
∫ 2
0
π(x^4 )dx
=π
∫ 2
0
(8x−x^4 )dx=π
[
8 x^2
2
−
x^5
5
] 2
0
=π
[(
16 −
32
5
)
−(0)
]
=9.6πcubic units
Now try the following exercise.
Exercise 150 Further problems on volumes

1. The curvexy=3 is revolved one revolution
   about thex-axis between the limitsx=2 and
   x=3. Determine the volume of the solid
   produced. [1.5πcubic units]


2. The area between

y

x^2

=1 andy+x^2 =8is

rotated 360◦about thex-axis. Find the vol-

ume produced. [170^23 πcubic units]

3. The curvey= 2 x^2 +3 is rotated about (a) the
   x-axis between the limitsx=0 andx=3,
   and (b) they-axis, between the same limits.
   Determine the volume generated in each case.
   [(a) 329.4π(b) 81π]


4. The profile of a rotor blade is bounded by the
   linesx= 0 .2,y= 2 x,y=e−x,x=1 and the
   x-axis. The blade thicknesstvaries linearly
   withxand is given by:t=(1. 1 −x)K, where
   K is a constant.
   (a) Sketch the rotor blade, labelling the
   limits.
   (b) Determine, using an iterative method, the
   value ofx, correct to 3 decimal places,
   where 2x=e−x
   (c) Calculate the cross-sectional area of the
   blade, correct to 3 decimal places.
   (d) Calculate the volume of the blade in terms
   of K, correct to 3 decimal places.
   [(b) 0.352 (c) 0.419 square units
   (d) 0.222 K]

38.5 Centroids

Alaminais a thin flat sheet having uniform thick-

ness. Thecentre of gravityof a lamina is the point